The name of the group to which a molecule belongs is determined
by the symmetry elements it possesses. Grouping together molecules
with the same symmetry elements will automatically group together
molecules of the same shape eg CX4 where X = H, F,
Cl, Br or I; as long as the four X groups are identical, these
molecules will all fall into the same point group. The various
point groups are as follows:
C1: Molecules which possess only the identity
operation. eg CBrClFI , a carbon atom with four different halogens
substituted onto it:
Ci: Molecules with the identity and
inversion alone. This is a relatively uncommon point group to
encounter in symmetry studies.e.g
Cs: Molecules with the identity and a mirror
plane alone. Again, this is not a commonly encountered point
Cn: Molecules with the identity and a Cn
axis alone. Another of the rarely-met point groups. A simple
example is hydrogen peroxide, H2O2 , which
belongs to C2:
Cnv: Molecules with the identity,
a Cn axis, and n vertical mirror planes, σv.
Commonly encountered examples are water, which belongs to C2v
, and ammonia, which belongs to C3v. A special subset
of this group is the group C∞v.
The group notation implies that all rotations around the axis
and reflections in a plane containing the axis are symmetry
operations. The group C∞v
consists, therefore, of all linear molecules that do not have
a centre of inversion, for example heteronuclear diatomics and
the linear OCS molecule
Cnh: Molecules with the identity,
a Cn principle axis, and a horizontal mirror plane.
e.g. B(OH)3 , in the correct conformation belongs
Dn: Molecules with the identity, an
n-fold principle axis and n C2 axes perpendicular
to the Cn axis.
Dnh: Molecules with the identity,
an n-fold principle axis, n perpendicular C2 axes,
and a horizontal mirror plane. Planar trigonal molecules
such as the BX3 series belong to D3h ,
while benzene belongs to the group D6h. There is
a subset of this group, D∞h
, which contains all linear molecules which do have a centre
of inversion, for example homonuclear diatomics and ethyne (C2H2).
Dnd: Molecules with the identity,
an n-fold principle axis, n C2 axes perpendicular
to the Cn axis, and n dihedral mirror planes (vertical
mirror planes which bisect the C2 axes). Allene,
H2C=C=CH2 , belongs to D2d:
Sn: Molecules that do not classify
into one of the above groups, but do possess an Sn
axis, belong to the group Sn. Molecules in this group
with n > 4 are very rare. Note there is no S2
group, as the S2 operation is equivalent to an inversion.
Thus a molecule that would appear to belong to S2
is classified as Ci.
Though there are examples of molecules available for all the
above groups, naturally some will be encountered more often
than others. Those likely to be encountered most often in studies
of symmetry are Cnv , Dnh
and Dnd. However, one should always
be alert to the existence of the other groups, and the possibility
that a molecule might classify into one of them instead.
The cubic groups, T, Td , Th , O,
Oh , I , Ih: These groups all have
high symmetry. They are distinguished from the above groups
by having no principle axis, but multiple equivalent Cn
axes with the highest number of n.
The group Td is the group of the regular tetrahedron
(so would include molecules such as methane, CH4).
The group T contains objects with the rotational symmetry of
a tetrahedron but none of its mirror planes, while Th
is based upon the T group but contains an inversion centre.
Oh is the group of a regular octahedron (so would
contain molecules such as SF6). O has the rotational
symmetry of the octahedron but no mirror planes. The icosahedral
group Ih has the symmetry of an icosahedron, and
contains the molecule C60, as well as some of the
more complex hydrides of boron.
In practice, only the groups Td , Oh ,
and Ih are likely to be encountered with any frequency.
The full rotation group, R3. Contains an
infinite number of rotational axes with all possible values
of n. This is the point group of a sphere or an atom.
No molecules belong to this group.
It should be noted that the presence of certain symmetry elements
automatically indicates the presence of others. For example,
an S2 axis implies the presence of an inversion centre,
as the two operations are equivalent. A C4 axis must
also be a C2 axis, as the C2 rotation
is equivalent to two successive C4 rotations. Numerous
similar examples exist.
Classification of molecules into their point groups is usually
achieved with some kind of key or flow diagram which must be
followed through answering questions about the symmetry elements
an object or molecule possesses, until a classification is made.
There are various different charts available for this purpose,
an example is given on the next page.